homework homework assignment #25 read section 4.5 page 243, exercises: 1 – 57 (eoo) rogawski...

Homework

Homework Assignment #25 Read Section 4.5 Page 243, Exercises: 1 – 57 (EOO)

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, 431. Match the graphs in Figure 12 with the description.

(b) (c) (a) (d)


a . 0 for all f x x c . 0 for all f x x

b . goes from + tof x d . goes from to +f x

Homework, Page 2435. If Figure 14 is the graph of the derivative of f ′(x), where do the points of inflection of f (x) occur, and on which interval is f (x) concave down?

Points of inflection

occur at a. and b.

the graph of f (x) is

concave down on

[d, f].


Homework, Page 243Determine the intervals on which the function is concave up or down and find the points of inflection.


9. 2cosy x x

2cos 1 2sin sin

Graph is concave up on 2 , 2 1

Graph is concave down on 2 1 ,2

Points of inflection at

y x x y x y x

n n

n n

x n

y " = sin x



2

113.

3y

x

2 22 2 2

2 22 2 2 2 2

2 42 2

2 2 22

3 3 3 32 2 2 2

1 21 2

3 3 3

3 2 2 2 3 2 2 3 8 3

3 3

2 3 8 6 1 6 1 16 6

3 3 3 3

x xy y

x x x

x x x x x x xy

x x

x x x x xx

x x x x

Homework, Page 24313. Continued

The function is concave up on (–∞, –1) and (1, ∞) and concave down on (–1, 1). Points of inflection are at

x = {–1, 1}.


32

1 16 1 1

3

xx xy

yx


The function is concave up on (–∞, –4.236) and (0.236, ∞) and concave down on (–4.236, 0.236). Points of inflection are at x = {–4.236, 0.236}.


217. 3 xy x e

2 2 2

2 2

3 3 2 2 3

2 3 2 2 4 1

4.236 0.2364.236 0.236

x x x x

x x x

x

y x e y x e e x x x e

y x x e e x x x e

xy x x e

y

Homework, Page 24321. The growth of a sunflower during its first 100 days is modeled well by the logistic curve y = h (t) in Figure 15. Estimate the growth rate at the point of inflection and explain its significance. Then make a rough sketch of the first and second derivatives of h (t).

The growth rate at the point of inflection appears to be about 7 cm/day. It is the greatest rate of growth as the second derivative goes from + to – at that point.


Homework, Page 243


x

y

20 40 60 80 100

y 'y '

Homework, Page 243Find the critical points of f (x) and use the Second Derivative Test to determine whether each corresponds to a local minimum or maximum.


4 3 225. 3 8 6f x x x x

3 2 2

2 2

22

2

2

12 24 12 12 2 1

36 48 12 12 3 4 1

0 12 2 1 12 1 0,1

0 12 3 0 4 0 1 12 Local minimum

1 12 3 1 4 1 1 0 Neither min nor max

f x x x x x x x

f x x x x x

f x x x x x x x

f

f



129.

cos 2f x

x

2 2

2

4

2 2

4

2

3

1 sin sin

cos 2 cos 2

cos 2 cos sin 2 cos 2 sin

cos 2

cos cos 2 2sin cos 2

cos 2

cos cos 2 2sin

cos 2

x xf x

x x

x x x x xf x

x

x x x x

x

x x x

x



2

3 3

2

3 3

0

cos0 cos0 2 2sin 0 1 1 2 2 00 0

cos0 2 1 2

cos cos 2 2sin 1 1 2 2 00

cos 2 1 2

Maximum at 2 1 , minimum at 2

f x x n

f

f

x n x n



2

33. xf x xe

2 2 2

2 2 2

2

2

2

2 3

2

3

3

2 1 1 2

1 2 2 4 4 6

10 1 2 0

2

1 1 14 6 0 Minimum

2 2 2

1 1 14 6 0 Maximum

2 2 2

x x x

x x x

x

x

f x xe x e x e

f x x e x e x x x e

f x x x

f e

f e

Homework, Page 243Find the intervals on which f is concave up or down, the points of inflection, and the critical points and determine whether each critical point corresponds to a local maximum or minimum (or neither).


3 237. 2f x x x x

2

2

3 4 1 6 4

13 4 1 0 ,1 Critical points

3

22 36 4 03

f x x x f x x

x x x

xx x

f



10 Local Maximum

3

1 0 Local Minimum

2 2Concave down on , ,Concave up on ,

3 3

2Inflection point at

3

f

f

x



12 241. f x x x

312 2

12

32

1 12 2

2 41

2 0 0.3972

12 0

4 is concave up on its domain, there are no

points of inflection and the critical point is a

local minimum

f x x x f x x

x x x

x

f



45. sin for 0 2f

1 cos sin

1 cos 0 sin 0

is concave down on 0,

and concave up on ,2 with inflection point

at . There are no extrema.

f f

f

ff

n



349. cos for

2 2xf x e x

sin cos sin cos

cos sin sin cos

2sin

30 sin cos cos sin ,

4 4

0 2sin 0, Inflection points at 0,

x x x

x x

x

x

x

f x e x x e e x x

f x e x x x x e

xe

f x e x x x x x

f x xe x x



0

Concave up on 0,

3Concave down on ,0 and ,

2 2

Relative maximum at 4

3Relative minimum at

4

x

f x

x

x

Homework, Page 24352. Water is pumped into a sphere at a variable rate in such a way that the water level rises at a constant rate c. Let V (t) be the volume at time t. Sketch the graph of V (t). Where does the point of inflection occur?


x

y

The point of inflection occurs at the time when the height of the water equals the radius of the tank.

x

y

Homework, Page 243Sketch the graph of a function satisfying the given condition.


57. i. 0 for all , and

ii. 0 for 0 and 0 for 0

f x x

f x x f x x

x

y


Jon Rogawski

Calculus, ETFirst Edition

Chapter 4: Applications of the DerivativeSection 4.5: Graph Sketching and

Asymptotes


Graphs of functions that are at least twice differentiable are made up of segments shown in Figure 1.

The keys to hand sketching are finding the transition points and selecting the correct curve shape.


The text uses solid dots to indicate local extrema and solidsquares to indicate transition points, as shown in Figure 2.


3 21 1In sketching the graph of 2 3,it is

3 2necessary first to find the critical points by setting 0.

The table below shows finding the sign of in the

intervals between the critical points.

y x x x

f

f


3 21 1In sketching the graph of 2 3,it is

3 2next necessary to find where 0. The table below

shows finding the sign of in the intervals between

the zeroes.

y x x x

f

f


Figure 4 shows the sign combinations of f ′ and f ″ for 3 21 1

2 3. 3 2

y x x x

Another way to represent this information is:

3 2 21 1 2 3 2 2 1

3 212 1 0, 1,2 ; 0, 2

11 22

y x x x y x x x x

y x y x y x

x

y

y


Calculating the values of f at the critical points and point ofinflection gives us sufficient information to sketch the graph.


23 2 2

2 2

12 24 12 12 2 1 12 1

36 48 12 12 3 4 1 12 3 1 1

10 13

f x x x x x x x x x

f x x x x x x x

x

f x

f x


1 1 5sin 0 sin

2 2 6, 6

cos 0 25

6 2 6

f x x x x

f x x x

x

f x

f x

Example, Page 256Sketch the graph of the function. Indicate the transition points.


314. 3 5y x x

x

y

Example, Page 256Sketch the graph of the function. Indicate the transition points.


34. 4 3lny x x x

x

y


Asymptotic behavior refers to the behavior of a function as either x orf (x) approaches ±∞. A horizontal line y = L is called a horizontal asymptote if either of the following exists:

lim or limx x

f x L f x L

Similarly, a vertical line y = L is called a vertical asymptote if eitherof the following exists:

lim or limx L x L

f x f x

Example, Page 256Sketch the graph over the given interval. Indicate the transition points.


42. sin , 0,2y x x

x

y

Example, Page 256Sketch the graph over the given interval. Indicate the transition points.


242. 2sin cos , 0,2y x x

x

y

Example, Page 256Calculate the following limits.


2

2

3 2052. lim

4 9x

x x

x

Example, Page 256Calculate the limit.


1

4 3

4 966. lim

3 2x

x

x

Example, Page 256Sketch the graph of the function. Indicate the asymptotes, local extrema, and points of inflection.


22

1 186.

2y

x x

x

y

Homework

Homework Assignment #26 Read Section 4.6 Page 256, Exercises: 1 – 89 (EOO)


homework homework assignment #25 read section 4.5 page 243, exercises: 1 – 57 (eoo) rogawski...

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